A simple case which falls in this regime is when \(x_{n}/n\)  converges to some limit belonging to the interval \((i_{\xi }, \rho G^{\prime }(\rho )/G(\rho ))\) . In this case, \(b_{n}\)  converges to a limit \(b \in (0, \rho )\) , and the generating function \(G\)  is very regular at this point, so one can readily adapt classical proofs of the local limit theorem in the Gaussian regime to the walk \(S^{(b_{n})}_{n}\)  and use (REF ).
This case is indeed known, see e.g. Borovkov & Borovkov [1]} and is included in the next theorem just for completeness. However, to the best of our knowledge the two other cases are new.
Note that no regularity assumptions are made on \(G\)  in the first two cases. For the last one, we recall from e.g. [2]}) that \(G\)  is said to be \(\Delta \) -analytic when there exist a radius \(R>\rho \)  and an angle \(\phi \in (0,\pi /2)\)  such that \(G\)  is analytic on the domain \( \lbrace z : |z|<R, z \ne \rho , |\arg (z-1)|>\phi \rbrace \) .
